unit4 frq3

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Sotunoss 2012 AP® STATISTICS FREE-RESPONSE QUESTIONS STATISTICS SECTION I Part A Questions 1-5 Spend about 65 minutes on this part of the exam. Percent of Section II score—75 Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 1. The scatterplot below displays the price in doltars and quality rating for 14 different sewing machines. ® 80 ° & clen ,5\ 2 70 ° /3?\ pourT 5 . N J & B2 6012 *e g 50 p ® 40f—= . 0 $500 $1,000 $1,500 $2000 $2,500 e 15 o canvede pon-Ineal assoc Price (dollars) )y . blw FY\LQ & ziw\\l{\a ra\%\rxs . (2) Describe the nature of the association between price and quality rating for the sewing machines. (b) One of the 14 sewing machines substantially affects the appropriateness of using a linear regression model to predict quality rating based on price. Report the approximate price and quality rating of that machine and explain your choice. 2 $3200 3 65 ¢ wl\'M) (c) Chris is interested in buying one of the 14 sewing machines. He will consider buying only those machines for which there is no other machine that ha§ both higher quality and lower price. On the scatterplot reproduced below, circle all data points corresponding to machines that Chris will consider buying. 80 o) - af ° g 70 .z\ 60 [] (] g & < 50 . ° 40 . 0 $500 $1,000 $1,500 $2,000 $2,500 Price (dollats) © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -6-

2012 AP® STATISTICS FREE-RESPONSE QUESTIONS 2. A charity fundraiser has a Spin the Pointer game that uses a spinner like the one illustrated in the figure below. A donation of $2 is required to play the game. For each $2 donation, a player spins the pointer once and receives the amount of money indicated in the sector where the pointer lands on the wheel. The spinner has an equal probability of landing in each of the 10 sectors. (a) Let X represent the net contribution to the chatity when oné person plays the game once. Complete the table for the probability distrbution of X. Conin buhon = $2 — (A///N'\’/\SJ P $2 $1 38 P@) o b .3 o | (b) What is the expected value of the net contribution to the charity for one play of the game? (c) The charity would like to receive a net contribution of $500 from this game. What is the fewest number of times the game must be played for the expected value of the net contribution to be at least $500 ? (d) Based on last year’s event, the charity anticipates that the Spin the Pointer game will be played 1,000 times. The charity would like to know the probability of obtaining a net contribution of at least $500 in 1,000 plays of the game. The mean and standard dev1auon of the net contribution to the charity i 92 lays of the game are $700 and $92.79, respectively. Use the normal distribution to approximate the probability that the charity Would obtain a net contribution of at least $500 in 1,000 plays of the game. b) Swptdys 2(.6)F ‘('35““8(")@ & 22 iy as—s IS 4;;;@ Pl e = SUD('?OD == 2,155 92,79 P(R7R.150) (T84 Y © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -7

2012 AP® STATISTICS FREE-RESPONSE QUESTIONS 3. Independent random samples of 500 households were taken from a large metropolitan area in the United States 6 1he years 1950 and 2000. Histograms of household size (number of people in a household) for the years are shown below. \o\rf)e enovo ro /,SD‘“O v Number of Households Number of Households 120 601 40 Household Size in 1950 o) Lower center in 2000 3-% Peopla, vs, ()aop\e, in V85D Less varmaton ia 000 beco\vse, of Kvge clugler -5 PeoP\Q vs . A peap(e, in 1950, frore exdve rre Values (\o+> in Oso then 2.000. Q,OH\ af skamﬂ-g h‘gkﬂ but more so o 3000, ble less yauseal 4o herve Sveh high numbergof pesple (o) P atleasT loX vapu s W16 6 8 10 Household Size in 2000 (a) Compare the distributions of household size in the metropolitan area for the years 1950 and 2000. (b) A researcher wants to use these data to construct a confidence interval to estimate the change in mean household size in the metropolitan area from the year 1950 to the year 2000. State the conditions for using a two-sample f-procedure, and explain whether the conditions for inference are met. © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -8-

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2012 AP® STATISTICS FREE-RESPONSE QUESTIONS 4. A survey organization conducted telephone interviews in December 2008 in which 1,009 randomly selected adults in the United States responded to the following question. ‘%’E\ At the present time, do you think television commercials are an effective way to promote a new product? I - car e, . Of tl@’lyfi@ adults surveyed, 676 responded@In December@m, 622 of 1,020 randonily selected adults in the United States had responded “yes” to the §ame question. Do the data provide convincing evidence that the proportion of adults in the United States who would respond “yes” to the question changed from December 2007 to December 2008 ? % e rop who wowld Sevyy @ Ho: ?,1()7, (' ‘\‘:)ei" in 200§ Ha P) #Po Py e peoP who wold Sab”\?,e; in Qo7 (D sesY soys so A 164630 = 2 ~ L 6397 l‘nde/e mdor\’f So\wfbs v . o@\PeaL \ob‘\m—;fi_ % .6 Y\/P\= Qobﬂ('(’?f‘-}) 2V’ P ) a02a(.3603)2 10 pop ot LoosT 1OX. S5 V7 @ "“? — a%"““""":_ﬂwzx‘ , 6700~ ,6098 ] ) “Je T o 2,820 Z \[!é\ (1 /?) '(J‘\’; + %‘“L \}’,25‘3'7 (, ‘3(,03) b 4 MJMW“ £ é fo09 jos.o @ TNt e p(H)>a8a00) o P (M\,Z\’A ~2.%6% 2,§9-9-6 @ Q&‘\\ec{' Ho uc .o04g .05, There s em’ufib\ erxdence o ; o/ / Sw%eg4’ Hot dhe QWFO("%{\ of US adulh who rpfl’)o(\oL 9% hes C(f\oﬂr\s?fo(- ’(\\OKY\ De ¢ 2.0077 “(0 Oee oo, © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -0-

2012 AP® STATISTICS FREE-RESPONSE QUESTIONS 5. A recent report stated that less than 35 percent of the adult residents in a certain city will be able to pass a physical fitness test. Consequently, the city’s Recreation Department is trying to convince the City Council to find more physical fitness programs. The council is facing budget constraints and is skeptical of the report. The council will fund more physical fitness programs only if the Recreation Department can provide convincing evidence that the report is true. The Recreation Department plans to collect data from a sample of 185 adult residents in the city. A test of significance will be conducted at a significance level of @ = 0.05 Tor the following hypotheses. Hy: p=035 H, : p <035 where p is the proportion of adult residents in the city who are able to pass the physical fitness test. (a) Describe what a Type II error would be in the context of the study, and also describe a consequence of making this type of error. (b) The Recreation Department recruits 185 adult residents w@lteer to fake the physical fitness test. The test is passed by 77 of the 185 volunteers, resulting in a p-valte 0 0:97 Tor the hypotheses stated above. If it was reasonable to conduct a test of significance for the hypotheses stated above using the data collected from the 185 volunteers, what would the p-value of 0.97 lead you to conclude? (c) Describe the primary flaw in the study described in part (b), and explain why it is a concern. . i oA the sy, o) 14 the fwe prop s less ¥han 35, buch v barl Ao regect the Fo we miss out on an oepov{w\‘d“b»h Fund- Lrvnoss proglems do address ot restdents not ke}r\s oble o pass e Glpess Hest, Conhinve 4o e pnore out of Q\'\@()Q Qr\a not Halnk Jhe Yesine P?oela will here (s & er@‘o\em ot Lness, o Ahod Here 1 clogoludely, no eddence Fhat b) T would Conclud - — the e prop of those who can pass o Aest is <. 35, This e ble a7 >.05. c) This is & vo\wd"arb response sample, Those who ave more ()\'\xoska\\g L) vl dend 4o ev\:)ob ‘e Q‘f\@l\@f\fifi of 4he fest and- went 4o \folwx‘\w‘ie«r} ond this Should pesult in an ove ceshimation of the ?a\(&\w\xz”‘f"( ok intere S+’v © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -10-

2012 AP® STATISTICS FREE-RESPONSE QUESTIONS STATISTICS SECTION II Part B Question 6 Spend about 25 minutes on this part of the exam. Percent of Section I score—25 Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 6. Two students at a large high school, Peter and Rania, wanted to estimate g, the mean number of soft drinks that a student at their school consumes in a week. A complete roster of the names and genders for the 2,000 students at their school was available. Peter selected a simple random sample of 100 students. Rania, knowing that 60 percent of the students at the school are Jemale, selecied a simple random sample oF 60 females a%bl an independent simple random sample ¢f 40 Ifizfi;j:Both asked all of the students in their samples how many soft drinks they typically consume in a week. &) +# students alphabeficaily 00DI-3.000. Block of & Y Jighs at ahme Inthe random # chart, (a) Describe a method Peter could have used 'to select a simple random sample of 100 students from the school. Select Rt oo pumbers, Ignaniny fe/eﬂa-fz & e pot bétueer 0o0i-3000. Peter and Rania conducted their studies as described. Peter used the sample mean X as a point estimator for 1. Rania used iovcm] = (0.6)§fcmalc + (0.4)§male as a point estimator for y, where }fema]e is the mean of the sample of 60 females and Xrmate is the mean of the sample of 40 males. Summary statistics for Petei’s data are shown in the table below. Variable N Mean Standard Deviation Number of . 1 soft drinks 100 532 4.13 \/ (b) Based on the summary statistics, calculate the estimated standard deviation of the sampling distribution (sometimes called the standard eiror) of Peter’s point estimator X. © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. “{1-

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2012 AP® STATISTICS FREE-RESPONSE QUESTIONS Summary statistics for Rania’s data are shown in the table below. Variable Gender N Mean Standard : Deviation Number of Female 60 2.90 1.80 soft drinks Male 40 745 2.22 Tronsforvathios) (c) Based on the summary statistics, calgéate the estigigted standard deviation of the sampling distribution of Rania’s point estimator Xoverall = (0.6)Xfemate + (0.4)Xmate. Ar () J +l’\€ \/o\r{a\ nee S |. e \* 220\ * (-6 J€6> + @L/} e Jomud + 0197136 & . © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. “12-

2012 AP® STATISTICS FREE-RESPONSE QUESTIONS A dotplot of Peter’s sample data is given below. scesese oescese : ° [ 3 ® L4 H $ o &8 8 8 S e ° * [ 3 L3 [3 3 [ o [ 3 0 2 4 6 8 10 12 14 16 Number of Soft Drinks Comparative dotplots of Rania’s sample data are given below. H s 2 | S ] A s o & & $ ¢ 8 3 2 & & o ¢ ¢ 8 ¢. % o % o e 9 © & s s ¢ Females —$—t—f—+—%—+ ¢+ 1777 T T 7 o 2 4 6 8 10 12 14 16 [ : H H ® & & @ [] IRy Males —p—p—p—p—T—4+—+—+—4+—+—4+—F—1t+—1"1 T o 2 4 6 8 10 12 14 16 Number of Soft Drinks (d) Using the dotplots above, explain why Rania’s point estimator has a smaller estimated standard deviation than the estimated standard deviation of Peter’s point estimator, Peters eshmate ollh not SF&»\'H(\Q 5\3 ,deeh, Theve wag a locq dklerences blw gender blc Rmales has a muek lower Ce/\V(‘(/\/\’/ 2-Y v, "&7—")) , FCW\A‘OJ Q\lsfi y\aé \QSS Verdahe n than YY\A{@S/ o\r\J l:ema(es were STOP mo e clugiecedd ayound the mean , Rania's method Jakes ENDOFEXAM [0 acct the dWfferences in conder, sha.ee)é-* Sprﬂca\,c/ ande Hhat (s (~"\‘\3 her eshmate 15 moce accuzde and hasa IO\/\\Q el S"\*A(\()A(T) de\/\\AhPS’V\ ' K es*’zl\M‘('CA- © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. 13-

unit4 frq3

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